Current methods of sending information through a channel which is bandwidth-limited are limited to a maximum error-free rate or channel capacity of C=W*log2(SN+1), where C is the channel capacity, W is the bandwidth, and SN is the signal to noise ratio. Equivalently, for a system with discrete amplitude levels. C=W*log2(N), where N is the number of available amplitude levels. The actual error-free transmission rate is of course smaller for discrete systems with noise. For more information see Shannon, C. E., “A Mathematical Method of Communication”, Reprinted from The Bell System Technical Journal, Vol. 27, July, October 1948, pp 379-423, 623-656. None of the modulation methods developed since the time of Shannon's landmark paper have superseded by more than a few percent the limits placed upon channel capacity presented in the above referenced paper. Embodiments of the present invention supersede the limits derived in the referenced paper by the expedient of sending overlapping symbols or pulses, which are then separated at the receiving end by deconvolution or an equivalent mathematical method. Shannon's derivation of the channel capacity makes the assumption that overlapping symbols or pulses are inseparable, a constraint which is obeyed by the current state of the art.
Previous applications of superresolution methods have been used to increase the amount of information discerned from an information source, but have not been directly applied to increase the transmission capacity of a communications channel. Engeler, et al. in U.S. Pat. Nos. 5,122,732 and 5,168,214 disclose application of superresolution methods to better determine the temporal spectrum of a signal, but did not use this to increase the information capacity or rate of the system. Stankwitz, et al. in U.S. Pat. No. 5,686,922 disclose a superresolution method to resolve beyond diffraction limits in radar. Abatzoglou, et al. in U.S. Pat. No. 5,748,507 disclose a method of using superresolution methods to improve resolution of frequency components of a signal. Cohen, et al. in U.S. Pat. No. 5,784,492 disclose use of a superresolution technique (the CLEAN algorithm) to compress a redundant data set. This method as presented applies to data compression and does not generate any gain in system transmission rate for non-redundant information. Gregory, et al. in U.S. Pat. No. 6,483,952 disclose a method for extracting additional information from an optical sensor system using superresolution methods, particularly with respect to spatial resolution information. Vaughan et. al., “Super-Resolution of Pulsed Multipath Channels for Delay Spread Characterization.” in IEEE Transactions on Communications vol. 47, no. 3, March 1999, pp. 343-347, utilizes superresolution methods to determine the effects of multipath propagation upon a signal.
Investigations of sending signals faster than Nyquist have been made, but none so far have utilized superresolution methods to cope with intersymbol interference. Landau in “Sampling, Data Transmission, and the Nyquist Rate” IEEE Proc. vol. 55, October 1967, pp. 1701-1706 performed calculations showing that for arbitrary signals, transmission beyond the Nyquist rate resulted in unstable measured signals. Later, in “On the Minimum Distance Problem for Faster-than-Nyquist Signaling”, IEEE Trans. Inf. Theory, vol. 34, no. 6 Nov. 1989, pp. 1420-1427, Landau and Mazo calculate the amount of intersymbol interference generated when binary valued signals are sent with very close spacing. Liveris et al. in “Exploiting faster-than-Nyquist signaling” IEEE Transactions on Communications vol. 51, no. 9, September 2003, pp. 1502-1511 also discuss the effect of intersymbol interference upon faster-than-Nyquist signaling, but do not discuss use of superresolution methods to deal with its effects. Zakarevicius et al. in “On the Speed Tolerance of Certain Classes of Data Transmission Systems” IEEE Transactions on Communications vol. 34, no. 8, August 1986, pp. 832-836 also analyze the effect of intersymbol interference upon faster-than-Nyquist signaling, but do not discuss the use of superresolution. Wang et al., in “Practically Realizable Digital Transmission Significantly Below the Nyquist Bandwidth” IEEE Transactions on Communications vol. 43, no. 2/3/4, February/March/April 1995, pp. 166-169, use digital pulses to transmit with a permissible amount of intersymbol interference, but do not use superresolution methods to detect the relative location of their digital pulses. All of the above methods of faster-than-Nyquist transmission rely upon establishing a maximum permissible level of intersymbol interference, which is less than an amount which can be reliably detected using standard means of pulse detection. This is different from the use of superresolution methods in that any amount of intersymbol interference can be removed by discernment with superresolution methods, subject to the constraints of resolution and noise.
Methods which send a sum of multiple parallel smaller bandwidth signals in place of a single signal occupying a bandwidth equal to the sum of the smaller bandwidths are well known (such as disclosed in U.S. Pat. No. 4,641,318 to Addeo), and may superficially resemble the method described herein, in that the convolution or correlation of a data signal with a small bandwidth signal results in a signal comprised of a sum of small bandwidth signals. However, none of these methods have anticipated using this type of method to achieve superresolution of the signal. Superresolution of the signal is required in order to transmit and receive information through a bandwidth-limited channel at a rate higher than C=W*log2(SN+1), which is the object of this invention. The primary differences which the method described herein possesses are that the extent of intersymbol interference induced is very much higher, and orthogonality between subsignals is not needed nor enforced. This is made possible by proper selection of the kernels and the constellation points at which the subsignals may be placed when the convolution is performed. The deconvolution thus places the subsignals at constellation points which can be distinguished in the presence of noise. In addition, accommodation is made for the case where subsignals sent at the same phase are summed and the intended data output must be distinguished by use of a subset-sum algorithm.